American Institute of Mathematical Sciences

Advanced
January  2013, 12(1): 519-546. doi: 10.3934/cpaa.2013.12.519

Numerical study of a family of dissipative KdV equations

Jean-Paul Chehab 1, and  Georges Sadaka 1,

1. 

LAMFA, UMR 6140, Université de Picardie Jules Verne, Pôle Scientifique, 33, rue Saint Leu, 80039 Amiens, France, France

Received  April 2011 Revised  December 2011 Published  September 2012

The weak damped and forced Korteweg-de Vries (KdV) equation on the 1d Torus have been analyzed by Ghidaglia[8, 9], Goubet[10, 11], Rosa and Cabral [3] where asymptotic regularization e ects have been proven and observed numerically. In this work, we consider a family of dampings that can be even weaker, particularly it can dissipate very few the high frequencies. We give numerical evidences that point out dissipation of energy, regularization e ect and the presence of special solutions that characterize a non trivial dynamics (steady states, time periodic solutions).
Keywords: Korteweg-de Vries equations, long time behavior, damping.
Mathematics Subject Classification: 35B10, 35B40, 35Q53, 65M12, 65M70.
Citation: Jean-Paul Chehab, Georges Sadaka. Numerical study of a family of dissipative KdV equations. Communications on Pure & Applied Analysis, 2013, 12 (1) : 519-546. doi: 10.3934/cpaa.2013.12.519
References:
[1]

M. Abounouh, H. Al Moatassime, J-P. Chehab, S. Dumont and O. Goubet, Discrete schrodinger equations and dissipative dynamical systems,, Communications on Pure and Applied Analysis, 7 (2008), 211.   Google Scholar

[2]

M. Abounouh, H. Al Moatassime, C. Calgaro and J-P. Chehab, A numerical scheme for the long time simulation of a forced damped KdV equation,, in preparation., ().   Google Scholar

[3]

M. Cabral and R. Rosa, Chaos for a damped and forced KdV equation,, Phys. D, 192 (2004), 265.  doi: 10.1016/j.physd.2004.01.023.  Google Scholar

[4]

C. Calgaro, J.-P. Chehab, J. Laminie and E. Zahrouni, Schémas multiniveaux pour les équations d'ondes,, (French) [Multilevel schemes for waves equations], 27 (2009), 180.  doi: 10.1051/proc/2009027.  Google Scholar

[5]

J.-P. Chehab and B. Costa, Time explicit schemes and spatial finite differences splittings,, Journal of Scientific Computing, 20 (2004), 159.   Google Scholar

[6]

J.-P. Chehab and G. Sadaka, Numerical study of a family of dissipative KdV equations,, preprint, (2011).   Google Scholar

[7]

D. Dutykh, Modélisation mathématique des Tsunamis,, (French) [Mathematical modeling of Tsunamis], (2007).   Google Scholar

[8]

J-M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time,, J. Diff. Eq., 74 (1988), 369.  doi: 10.1016/0022-0396(88)90010-1.  Google Scholar

[9]

J-M. Ghidaglia, A note on the strong convergence towards attractors for damped forced KdV equations,, J. Diff. Eq., 110 (1994), 356.  doi: 10.1006/jdeq.1994.1071.  Google Scholar

[10]

O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations,, Discrete Contin. Dynam. Systems, 6 (2000), 625.   Google Scholar

[11]

O. Goubet and R. Rosa, Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line,, J. Differential Equations, 185 (2002), 25.  doi: 10.1006/jdeq.2001.4163.  Google Scholar

[12]

D. Gottlieb and S. A. Orszag, "Numerical Analysis of Spectral Methods: Theory and Applications,", SIAM Philadelphia, (1977).  doi: 10.1115/1.3424477.  Google Scholar

[13]

S. Mallat, "A Wavelet Tour of Signal Processing,", Academic press, (1998).   Google Scholar

[14]

A. Miranville and R. Temam, "Mathematical Modeling in Continuum Mechanics,", Cambridge University Press, (2005).   Google Scholar

[15]

L. Molinet and S. Vento, Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the periodic case,, preprint, ().   Google Scholar

[16]

E. Ott and R. N. Sudan, Nonlinear theory of ion acoustic wave with Landau damping,, Physics of fluids, 12 (1969), 2388.  doi: 10.1063/1.1692358.  Google Scholar

[17]

E. Ott and R. N. Sudan, Damping of solitary waves,, Physics of fluids, 13 (1970), 1432.  doi: 10.1063/1.1693097.  Google Scholar

[18]

A. Duràn and J. M. Sanz-Serna, The numerical integration of a relative equilibrium solutions. the nonlinear schrodinger equation,, IMA J. Num. Anal., 20 (2000), 235.  doi: 10.1093/imanum/20.2.235.  Google Scholar

[19]

R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics,", 2$^{nd}$ edition, (1997).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[20]

S. Vento, Global well-posedness for dissipative Korteweg-de Vries equations,, Funkcialaj Ekvacioj, 54 (2011), 119.  doi: 10.1619/fesi.54.119.  Google Scholar

[21]

S. Vento, Asymptotic behavior for dissipative Korteweg-de Vrie equations,, Asymptot. Anal., 68 (2010), 155.   Google Scholar

show all references

References:
[1]

M. Abounouh, H. Al Moatassime, J-P. Chehab, S. Dumont and O. Goubet, Discrete schrodinger equations and dissipative dynamical systems,, Communications on Pure and Applied Analysis, 7 (2008), 211.   Google Scholar

[2]

M. Abounouh, H. Al Moatassime, C. Calgaro and J-P. Chehab, A numerical scheme for the long time simulation of a forced damped KdV equation,, in preparation., ().   Google Scholar

[3]

M. Cabral and R. Rosa, Chaos for a damped and forced KdV equation,, Phys. D, 192 (2004), 265.  doi: 10.1016/j.physd.2004.01.023.  Google Scholar

[4]

C. Calgaro, J.-P. Chehab, J. Laminie and E. Zahrouni, Schémas multiniveaux pour les équations d'ondes,, (French) [Multilevel schemes for waves equations], 27 (2009), 180.  doi: 10.1051/proc/2009027.  Google Scholar

[5]

J.-P. Chehab and B. Costa, Time explicit schemes and spatial finite differences splittings,, Journal of Scientific Computing, 20 (2004), 159.   Google Scholar

[6]

J.-P. Chehab and G. Sadaka, Numerical study of a family of dissipative KdV equations,, preprint, (2011).   Google Scholar

[7]

D. Dutykh, Modélisation mathématique des Tsunamis,, (French) [Mathematical modeling of Tsunamis], (2007).   Google Scholar

[8]

J-M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time,, J. Diff. Eq., 74 (1988), 369.  doi: 10.1016/0022-0396(88)90010-1.  Google Scholar

[9]

J-M. Ghidaglia, A note on the strong convergence towards attractors for damped forced KdV equations,, J. Diff. Eq., 110 (1994), 356.  doi: 10.1006/jdeq.1994.1071.  Google Scholar

[10]

O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations,, Discrete Contin. Dynam. Systems, 6 (2000), 625.   Google Scholar

[11]

O. Goubet and R. Rosa, Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line,, J. Differential Equations, 185 (2002), 25.  doi: 10.1006/jdeq.2001.4163.  Google Scholar

[12]

D. Gottlieb and S. A. Orszag, "Numerical Analysis of Spectral Methods: Theory and Applications,", SIAM Philadelphia, (1977).  doi: 10.1115/1.3424477.  Google Scholar

[13]

S. Mallat, "A Wavelet Tour of Signal Processing,", Academic press, (1998).   Google Scholar

[14]

A. Miranville and R. Temam, "Mathematical Modeling in Continuum Mechanics,", Cambridge University Press, (2005).   Google Scholar

[15]

L. Molinet and S. Vento, Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the periodic case,, preprint, ().   Google Scholar

[16]

E. Ott and R. N. Sudan, Nonlinear theory of ion acoustic wave with Landau damping,, Physics of fluids, 12 (1969), 2388.  doi: 10.1063/1.1692358.  Google Scholar

[17]

E. Ott and R. N. Sudan, Damping of solitary waves,, Physics of fluids, 13 (1970), 1432.  doi: 10.1063/1.1693097.  Google Scholar

[18]

A. Duràn and J. M. Sanz-Serna, The numerical integration of a relative equilibrium solutions. the nonlinear schrodinger equation,, IMA J. Num. Anal., 20 (2000), 235.  doi: 10.1093/imanum/20.2.235.  Google Scholar

[19]

R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics,", 2$^{nd}$ edition, (1997).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[20]

S. Vento, Global well-posedness for dissipative Korteweg-de Vries equations,, Funkcialaj Ekvacioj, 54 (2011), 119.  doi: 10.1619/fesi.54.119.  Google Scholar

[21]

S. Vento, Asymptotic behavior for dissipative Korteweg-de Vrie equations,, Asymptot. Anal., 68 (2010), 155.   Google Scholar

[1]

Pierre Garnier. Damping to prevent the blow-up of the korteweg-de vries equation. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1455-1470. doi: 10.3934/cpaa.2017069
[2]

Brian Pigott. Polynomial-in-time upper bounds for the orbital instability of subcritical generalized Korteweg-de Vries equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 389-418. doi: 10.3934/cpaa.2014.13.389
[3]

Belkacem Said-Houari. Long-time behavior of solutions of the generalized Korteweg--de Vries equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 245-252. doi: 10.3934/dcdsb.2016.21.245
[4]

Eduardo Cerpa. Control of a Korteweg-de Vries equation: A tutorial. Mathematical Control & Related Fields, 2014, 4 (1) : 45-99. doi: 10.3934/mcrf.2014.4.45
[5]

M. Agrotis, S. Lafortune, P.G. Kevrekidis. On a discrete version of the Korteweg-De Vries equation. Conference Publications, 2005, 2005 (Special) : 22-29. doi: 10.3934/proc.2005.2005.22
[6]

Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Global stabilization of a coupled system of two generalized Korteweg-de Vries type equations posed on a finite domain. Mathematical Control & Related Fields, 2011, 1 (3) : 353-389. doi: 10.3934/mcrf.2011.1.353
[7]

Netra Khanal, Ramjee Sharma, Jiahong Wu, Juan-Ming Yuan. A dual-Petrov-Galerkin method for extended fifth-order Korteweg-de Vries type equations. Conference Publications, 2009, 2009 (Special) : 442-450. doi: 10.3934/proc.2009.2009.442
[8]

Olivier Goubet. Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 625-644. doi: 10.3934/dcds.2000.6.625
[9]

Guolian Wang, Boling Guo. Stochastic Korteweg-de Vries equation driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5255-5272. doi: 10.3934/dcds.2015.35.5255
[10]

Muhammad Usman, Bing-Yu Zhang. Forced oscillations of the Korteweg-de Vries equation on a bounded domain and their stability. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1509-1523. doi: 10.3934/dcds.2010.26.1509
[11]

Eduardo Cerpa, Emmanuelle Crépeau. Rapid exponential stabilization for a linear Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 655-668. doi: 10.3934/dcdsb.2009.11.655
[12]

Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control & Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61
[13]

Ludovick Gagnon. Qualitative description of the particle trajectories for the N-solitons solution of the Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1489-1507. doi: 10.3934/dcds.2017061
[14]

Arnaud Debussche, Jacques Printems. Convergence of a semi-discrete scheme for the stochastic Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 761-781. doi: 10.3934/dcdsb.2006.6.761
[15]

Qifan Li. Local well-posedness for the periodic Korteweg-de Vries equation in analytic Gevrey classes. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1097-1109. doi: 10.3934/cpaa.2012.11.1097
[16]

Anne de Bouard, Eric Gautier. Exit problems related to the persistence of solitons for the Korteweg-de Vries equation with small noise. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 857-871. doi: 10.3934/dcds.2010.26.857
[17]

Shou-Fu Tian. Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval. Communications on Pure & Applied Analysis, 2018, 17 (3) : 923-957. doi: 10.3934/cpaa.2018046
[18]

Roberto A. Capistrano-Filho, Shuming Sun, Bing-Yu Zhang. General boundary value problems of the Korteweg-de Vries equation on a bounded domain. Mathematical Control & Related Fields, 2018, 8 (3&4) : 583-605. doi: 10.3934/mcrf.2018024
[19]

John P. Albert. A uniqueness result for 2-soliton solutions of the Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3635-3670. doi: 10.3934/dcds.2019149
[20]

Zhaosheng Feng, Yu Huang. Approximate solution of the Burgers-Korteweg-de Vries equation. Communications on Pure & Applied Analysis, 2007, 6 (2) : 429-440. doi: 10.3934/cpaa.2007.6.429

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences